Finiteness results for Abelian tree models

Equivariant tree models are statistical models used in the reconstruction of phylogenetic trees from genetic data. Here equivariant refers to a symmetry group imposed on the root distribution and on the transition matrices in the model. We prove that if that symmetry group is Abelian, then the Zariski closures of these models are defined by polynomial equations of bounded degree, independent of the tree. Moreover, we show that there exists a polynomial-time membership test for that Zariski closure. This generalises earlier results on tensors of bounded rank, which correspond to the case where the group is trivial, and implies a qualitative variant of a quantitative conjecture by Sturmfels and Sullivant in the case where the group and the alphabet coincide. Our proofs exploit the symmetries of an infinite-dimensional projective limit of Abelian star models.